This theorem follows from an inductive estimate of the fermionic seminorm of nested commutators of Hamiltonian terms, and will be formally proved in [sec:recurse]Section sec:recurse and [sec:path]Section sec:path. Note that in order to use the prior knowledge of the initial state, we have considered the fermionic seminorm \(\left\Vert \cdot \right\Vert _{\eta }\)  of Trotter error with respect to the \(\eta \) -electron manifold. This seminorm is closely related to other metrics used to quantify the impact of initial-state information to digital quantum simulation [1]}, [2]}, [3]}, [4]}; see [sec:prelimseminorm]Section sec:prelimseminorm for a detailed discussion. The resulting bound depends on the number of electrons \(\eta \) , as well as the spectral norm \(\left\Vert \tau \right\Vert \) , the max-norm \(\left\Vert \tau \right\Vert _{\max },\left\Vert \nu \right\Vert _{\max }\) , and the sparsity of interactions \(d\) , but there is no dependence on the total number of spin orbitals \(n\) . This improves over previous work [5]} where an explicit \(n\) -scaling seems unavoidable. Meanwhile, other prior estimates of the fermionic seminorm [3]} [4]} did not exploit commutativity of the Hamiltonian and would introduce an additional factor of \(\eta ^{p}\)  in the Trotter error bound. Our result thus improves the performance of digital quantum simulation by combining the initial-state information, the interaction sparsity, and commutativity of the Hamiltonian.
